Care and Value of Model Assumptions: Case study

Case study: how do features of nesting female horseshoe crabs influence the number of males found nearby?

Load the data. Here are the top six rows of 173 rows:

colour	spine	width	weight	n_male
2	3	28.3	3.05	8
3	3	26.0	2.60	4
3	3	25.6	2.15	0
4	2	21.0	1.85	0
2	3	29.0	3.00	1
1	2	25.0	2.30	3

Predictors: Colour; spine condition; carapace width; weight.

First, let’s see how carapace width influences the mean number of males nearby.

Data source: H. Jane Brockmann’s 1996 paper; found online here; another regression demo with this data is found here.

Approach 1: Estimate regression curve / model function locally

Preliminary questions

These questions are meant to check your understanding of local regression.

What is the estimated mean number of nearby males for nesting females having a carapace width of 32.5? Use the following methods, by hand.

1. kNN with \(k=3\).

2. Using a moving window with a radius of 2.4.

3. Using a kernel smoother with Gaussian kernel with variance 1.

4. Using local polynomials with a radius of 2.4 and a flat kernel, first with degree 1, then with degree 2.

Fit a smoother by eye

Optimize the loess fit by-eye. Just modify span, to keep things simple.

grid <- seq(min(crab$width), max(crab$width), length.out=100)
grid_df <- tibble(width = grid)
# FIT_MODEL_HERE
# PLOT_CURVE_HERE

What’s the error of this model? Training error is fine.

How well does this model answer our original question?

Approach 2: Linear Regression

Fit a linear regression model

Fit a linear regression model. What’s the error?

How well does this model answer our original question? Do you see a potential problem with this model? Are any assumptions of linear regression not true? Brainstorm ideas for how to deal with the problems.

Approach 3: Link Function

Fit a GLM. What’s the error?